I remember reading a short story on the internet where teleportation technology is invented. The main character decries the new technology and refuses to use it because it causes a gap in consciousness, and thus results in the death of the person who uses it.
He eventually has the realisation (he might have come to this realisation after a debate with an advocate of the teleportation technology, but I'm unsure of this) that if a gap in consciousness is fatal, he dies and is reborn every time he goes to sleep. He then tries to avoid sleeping for as long as he can before eventually succumbing.
I cannot remember what happens after he falls asleep.